Finite Group Databasebeta

← #27538899

Subgroups — #27538899

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284292C211yesmaximal, characteristic
2C2×#107×D53C211yesmaximal, characteristic, complemented
3#20276916C212yesmaximal, complemented
4#20276911C212yesmaximal, complemented
5#20276899C212yesmaximal, complemented
6#20276894C212yesmaximal, complemented
7#20276503C211yesmaximal, characteristic, complemented
8#20276501C211yesmaximal, characteristic, complemented
9#20276427C211yesmaximal, characteristic, complemented
10#20276325C211yesmaximal, characteristic, complemented
11C53×#2139C211yesmaximal, characteristic, complemented
12C22×C8×D53C2211yescharacteristic
13#16811130—21no—
14C2×C4⋊C4×D53—21no—
15C2×#13133157C2211yescharacteristic
16#16810861C2211yescharacteristic
17#16810861C2211yescharacteristic
18#16810856C2211yescharacteristic
19#16810856C2211yescharacteristic
20#16810849—21no—
21#16810849—21no—
22#16810661C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133150—21no—
25#107×D53—21no—
26#107×D53—21no—
27#107×D53C2212yes—
28#107×D53C2212yes—
29#16810619—21no—
30#16810451C2212yescomplemented
31#16810450C2212yescomplemented
32#16810449C2212yescomplemented
33#16810448C2212yescomplemented
34#16810427C2212yescomplemented
35#16810424C2212yescomplemented
36#16810402C2212yes—
37#16810402C2212yes—
38C2×#135×C53C2211yescharacteristic
39C2×M32×C53C2211yescharacteristic
40C2×#107×C53C2211yescharacteristic
41C53×#358C2212yescomplemented
42C53×#357C2212yescomplemented
43C22×C4×D53—41no—
44C22×C4×D53D411yescharacteristic
45C22×#88322D411yescharacteristic
46C2×C8×D53—22no—
47C2×C8×D53—22no—
48C2×C8×D53D411yescharacteristic
49C2×C8×D53D411yescharacteristic
50C2×C8×D53C2311yescharacteristic
51C2×C8×D53C2311yescharacteristic
52#13133198—41no—
53#13133193—21no—
54#13133193—21no—
55#13133188—21no—
56#13133188—21no—
57#13133185—41no—
58C4×C22⋊D53—41no—
59C4×C22⋊D53—41no—
60#13133176—21no—
61C2×C4⋊Dic53—21no—
62C2×#11719919—41no—
63C4⋊C4×D53—22no—
64C4⋊C4×D53—22no—
65C4⋊C4×D53—41no—
66C4⋊C4×D53—41no—
67#13133157—22no—
68C2×#11719908C2312yes—
69#13133151—21no—
70#13133151C2311yescharacteristic
71#13133151C2311yescharacteristic
72#13133151C2312yes—
73#13133150—22no—
74#13133150—22no—
75#13133149—21no—
76#13133141—21no—
77#13133141—21no—
78#13133138—21no—
79#13133138—21no—
80C2×#41×C53—21no—
81#135×C53—22no—
82#135×C53—22no—
83C2×Q16×C53C2311yescharacteristic
84C2×SD16×C53—21no—
85C2×D8×C53C2311yescharacteristic
86C22×C8×C53C2311yescharacteristic
87C2×C4⋊C4×C53—21no—
88M32×C53—22no—
89C2×C16×C53C2312yes—
90#107×C53—21no—
91#107×C53C2312yes—
92C23×D53D811yescharacteristic
93#41×C53—42no—
94#41×C53—42no—
95C2×Q8×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C2×C22⋊D53—41no—
102C2×C22⋊D53—41no—
103C22×Dic53—41no—
104C22×Dic53D811yescharacteristic
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—42no—
108C2×C4×D53—42no—
109C2×C4×D53—84no—
110C2×C4×D53D811yescharacteristic
111C2×C4×D53D811yescharacteristic
112C2×C4×D53C2×D411yescharacteristic
113C2×C4×D53C2×D411yescharacteristic
114Q16×C53—21no—
115Q16×C53—21no—
116SD16×C53—42no—
117D8×C53—21no—
118D8×C53—21no—
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53#4112yes—
121C2×C8×C53#4112yes—
122C2×C8×C53C2411yescharacteristic
123C4⋊C4×C53—22no—
124C4⋊C4×C53—41no—
125C22⋊Dic53—41no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128#11719921—41no—
129#11719921—41no—
130C4⋊Dic53—21no—
131C4⋊Dic53—21no—
132C4⋊Dic53—22no—
133C4⋊Dic53—41no—
134#11719919—42no—
135#11719919—42no—
136#11719919—41no—
137#11719919—41no—
138C4×Dic53—41no—
139C2×#88322—22no—
140C2×#88322C2×D411yescharacteristic
141C2×#88322C2×D411yescharacteristic
142C8×D53—22no—
143C8×D53—22no—
144C8×D53—21no—
145C8×D53—21no—
146C8×D53—21no—
147C8×D53—21no—
148C16×C53—22no—
149#11719908—22no—
150C23×C53—41no—
151C23×C53C2×D811yescharacteristic
152C22×D53—22no—
153C22×D53—22no—
154C22×D53—22no—
155C22×D53C2×D811yescharacteristic
156C22×D53C2×D811yescharacteristic
157Q8×C53—41no—
158Q8×C53—41no—
159D4×C53—41no—
160D4×C53—41no—
161D4×C53—82no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—82no—
165C2×C4×C53—82no—
166C2×C4×C53#12312yes—
167C2×C4×C53C2×D811yescharacteristic
168C2×C4×C53#13712yes—
169C2×C4×C53C22×D411yescharacteristic
170C22⋊D53—82no—
171C22⋊D53—82no—
172C2×Dic53—22no—
173C2×Dic53—42no—
174C2×Dic53—41no—
175C2×Dic53—41no—
176C2×Dic53—82no—
177C2×Dic53C2×D811yescharacteristic
178C2×Dic53C2×D811yescharacteristic
179C4×D53—21no—
180C4×D53—21no—
181C4×D53—21no—
182C4×D53—21no—
183C4×D53—22no—
184C4×D53—22no—
185C4×D53—84no—
186C4×D53—84no—
187C8×C53#12312yes—
188C8×C53C2×#4111yescharacteristic
189C8×C53C2×#4111yescharacteristic
190#88322—21no—
191#88322—21no—
192#33646—531nomaximal, Sylow
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53#48012yes—
199C22×C53C22×D811yescharacteristic
200C2×D53—28no—
201C2×D53—21no—
202C2×D53—21no—
203C2×D53—21no—
204C2×D53—21no—
205C2×D53#50912yes—
206C2×D53#51012yes—
207C4×C53—81no—
208C4×C53—82no—
209C4×C53#48012yes—
210C4×C53C2×#12311yescharacteristic
211C4×C53C2×#13711yescharacteristic
212Dic53—21no—
213Dic53—21no—
214Dic53—82no—
215Dic53—81no—
216Dic53—81no—
217#3050—531no—
218C22×#107—531no—
219#2236—532no—
220#2235—532no—
221#2233—532no—
222#2232—532no—
223#2159—531no—
224#2159—531no—
225#2139—531no—
226#2139D5311yescharacteristic, complemented
227#2112—531no—
228C2×C53—22no—
229C2×C53—161no—
230C2×C53C2×#48011yescharacteristic
231C2×C53C2×#50911yescharacteristic
232C2×C53C2×#51011yescharacteristic
233D53—22no—
234D53—22no—
235C2×#135C2×D5311yescharacteristic
236C23×C8—531no—
237#525—1061no—
238C22×C4⋊C4—1061no—
239C2×M32—531no—
240C2×M32C2×D5311yescharacteristic
241#467—531no—
242#467—531no—
243#466—531no—
244#466—531no—
245#465—1061no—
246#465—1061no—
247#427—531no—
248C2×#107—531no—
249C2×#107—532no—
250C2×#107—532no—
251C2×#107—1061no—
252C2×#107—1061no—
253C2×#107—1061no—
254C2×#107C2×D5311yescharacteristic
255#416—1061no—
256#367—532no—
257#367—532no—
258#366—532no—
259#366—532no—
260#358—532no—
261#358C2×D5312yescomplemented
262#357—532no—
263#357C2×D5312yescomplemented
264C22⋊C16—532no—
265C22⋊C16—532no—
266C53#3364611yescharacteristic, complemented, Sylow
267C2×#41—21no—
268C23×C4—531no—
269C23×C4—2121no—
270#135—22no—
271#135—22no—
272C2×Q16C22×D5311yescharacteristic
273C2×SD16—21no—
274C2×D8C22×D5311yescharacteristic
275C22×C8—531no—
276C22×C8—531no—
277C22×C8—531no—
278C22×C8—531no—
279C22×C8—531no—
280C22×C8—1062no—
281C22×C8—1062no—
282C22×C8C22×D5311yescharacteristic
283#123—2121no—
284C22⋊Q8—1061no—
285C22⋊Q8—1061no—
286#121—1061no—
287#121—1061no—
288#120—2121no—
289C4×D4—2121no—
290C4×D4—2121no—
291#117—1061no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—1062no—
294C2×C4⋊C4—1062no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—2121no—
297C2×C4⋊C4—2121no—
298C2×C4⋊C4—2121no—
299M32—22no—
300M32—1062no—
301C2×C16—532no—
302C2×C16C22×D5312yes—
303#107—21no—
304#107—532no—
305#107—531no—
306#107—531no—
307#107—1062no—
308#107—1062no—
309#107—1061no—
310#107C22×D5312yes—
311#106—1061no—
312#103—1061no—
313#103—1061no—
314#102—1061no—
315#102—1061no—
316C24—531no—
317#41—42no—
318#41—42no—
319C2×Q8—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4—2121no—
323C2×D4—2121no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—531no—
327C22×C4—531no—
328C22×C4—531no—
329C22×C4—531no—
330C22×C4—531no—
331C22×C4—1062no—
332C22×C4—1062no—
333C22×C4—4244no—
334C22×C4—2122no—
335C22×C4—2122no—
336C22×C4—2121no—
337C22×C4D4×D5311yescharacteristic
338Q16—21no—
339Q16—21no—
340SD16—42no—
341D8—21no—
342D8—21no—
343C2×C8—531no—
344C2×C8—531no—
345C2×C8—1062no—
346C2×C8—1062no—
347C2×C8—1062no—
348C2×C8—1061no—
349C2×C8—1061no—
350C2×C8—1061no—
351C2×C8—1061no—
352C2×C8#1171994512yes—
353C2×C8D4×D5311yescharacteristic
354C2×C8#1171994712yes—
355C2×C8C23×D5311yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—1061no—
359C4⋊C4—1061no—
360C4⋊C4—1062no—
361C4⋊C4—2122no—
362C4⋊C4—2122no—
363C4⋊C4—2121no—
364C4⋊C4—2121no—
365C4⋊C4—2121no—
366C22⋊C4—2121no—
367C22⋊C4—2121no—
368C22⋊C4—2121no—
369C22⋊C4—2121no—
370C22⋊C4—2121no—
371C42—2121no—
372C16—22no—
373C16—1062no—
374C23—41no—
375C23—531no—
376C23—531no—
377C23—1062no—
378C23—1062no—
379C23—1062no—
380C23D8×D5311yescharacteristic
381Q8—41no—
382Q8—41no—
383D4—41no—
384D4—41no—
385D4—82no—
386D4—4242no—
387D4—4242no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—531no—
393C2×C4—531no—
394C2×C4—1062no—
395C2×C4—1061no—
396C2×C4—1061no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1062no—
400C2×C4—1062no—
401C2×C4—4244no—
402C2×C4—4244no—
403C2×C4—4242no—
404C2×C4—2121no—
405C2×C4—2121no—
406C2×C4—2122no—
407C2×C4#1313319912yes—
408C2×C4D8×D5311yescharacteristic
409C2×C4#1313324612yes—
410C2×C4C2×D4×D5311yescharacteristic
411C8—1061no—
412C8—1061no—
413C8#1313319912yes—
414C8C2×#1171994711yescharacteristic
415C8C2×#1171994511yescharacteristic
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—532no—
422C22—532no—
423C22—1068no—
424C22—1061no—
425C22—1061no—
426C22—1061no—
427C22—1061no—
428C22#1681093512yes—
429C22C2×D8×D5311yescharacteristic
430C4—82no—
431C4—81no—
432C4—1061no—
433C4—1061no—
434C4—4242no—
435C4—4241no—
436C4—4241no—
437C4#1681093512yes—
438C4C2×#1313319911yescharacteristic
439C4C2×#1313324611yescharacteristic
440C2—22no—
441C2—161no—
442C2—1062no—
443C2—1062no—
444C2C2×#1681093511yescharacteristic
445C2#509×D5311yescharacteristic
446C2#510×D5311yescharacteristic
subgroupG/H
centerC22C2×D8×D53
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC53×#2139C2
socleC22×C53C22×D8
prime radical#20284292C2
hypercenter#2139D53
hypocenterC53#33646
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#107×D53C2semidirect
3. #20276916C2semidirect
4. #20276911C2semidirect
5. #20276899C2semidirect
6. #20276894C2semidirect
7. #20276503C2semidirect
8. #20276501C2semidirect
9. #20276427C2semidirect
10. #20276325C2semidirect
11. C53×#2139C2semidirect
30. #16810451C22semidirect
31. #16810450C22semidirect
32. #16810449C22semidirect
33. #16810448C22semidirect
34. #16810427C22semidirect
35. #16810424C22semidirect
41. C53×#358C22semidirect
42. C53×#357C22semidirect
226. #2139D53semidirect
261. #358C2×D53semidirect
263. #357C2×D53semidirect
266. C53#33646semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202842922
C2×#107×D532
#202769162
#202769112
#202768992
#202768942
#202765032
#202765012
#202764272
#202763252
C53×#21392
#3364653